Polynomial Equations and the Remainder Theorem

A polynomial equation of degree $n$ has the form $a_0x^n + a_1x^{n-1} + \cdots + a_{n-1}x + a_n = 0$, with $a_0 \neq 0$. Up to now you have solved linear and quadratic equations directly, but most equations that show up in real applications — in engineering, economics, or the sciences — have degree three or higher, and there is no simple formula for their roots in general. The theory of equations is the set of ideas that lets us say useful, precise things about the roots of any polynomial equation even when we cannot write those roots down explicitly.

The starting point is the Fundamental Theorem of Algebra: every polynomial equation of degree $n \ge 1$ with complex coefficients has at least one root in $\mathbb{C}$. Applying this repeatedly (peel off one root, divide it out, repeat) shows that a degree-$n$ polynomial equation has exactly $n$ roots in $\mathbb{C}$, provided we count a repeated root as many times as it repeats (its multiplicity). This is why we can always write

for some complex numbers $\alpha_1,\dots,\alpha_n$ — these are exactly the roots.

A second essential tool is the Remainder Theorem: if a polynomial $f(x)$ of degree $n>0$ is divided by $(x-a)$, the remainder is simply $f(a)$, i.e. $f(x) = (x-a)\,q(x) + f(a)$ for some quotient $q(x)$ of degree $n-1$. In particular, $a$ is a root of $f(x)=0$ exactly when $f(a)=0$, i.e. exactly when $(x-a)$ divides $f(x)$ with zero remainder. This single fact underlies almost every technique in this chapter: testing a candidate root, factoring out a known root, and building up the quotient step by step. …